Mister Exam

Lang:

Other calculators

How to use it?
Derivative of:
Derivative of x/(x+1)
Derivative of x/log(x)
Derivative of (x+1)/(x-1)
Derivative of (x^2-4)/x
Graphing y =:
x/(x-1)^2
Integral of d{x}:
x/(x-1)^2
Identical expressions
x/(x- one)^ two
x divide by (x minus 1) squared
x divide by (x minus one) to the power of two
x/(x-1)2
x/x-12
x/(x-1)²
x/(x-1) to the power of 2
x/x-1^2
x divide by (x-1)^2
Similar expressions
x/(x+1)^2

The derivative of the function/ x/(x-1)^2

Derivative of x/(x-1)^2

The solution

You have entered [src]

   x    
--------
       2
(x - 1)

$$\frac{x}{\left(x - 1\right)^{2}}$$

x/(x - 1)^2

Detail solution

Apply the quotient rule, which is:

$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$

$f{\left(x \right)} = x$ and $g{\left(x \right)} = \left(x - 1\right)^{2}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Apply the power rule: $x$ goes to $1$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = x - 1$ .
2. Apply the power rule: $u^{2}$ goes to $2 u$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x - 1\right)$ :
  1. Differentiate $x - 1$ term by term:
    1. The derivative of the constant $-1$ is zero.
    2. Apply the power rule: $x$ goes to $1$
    The result is: $1$
  The result of the chain rule is:
  
  $2 x - 2$
Now plug in to the quotient rule:

$\frac{- x \left(2 x - 2\right) + \left(x - 1\right)^{2}}{\left(x - 1\right)^{4}}$
Now simplify:

$- \frac{x + 1}{\left(x - 1\right)^{3}}$

The answer is:

$- \frac{x + 1}{\left(x - 1\right)^{3}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

   1       x*(2 - 2*x)
-------- + -----------
       2            4 
(x - 1)      (x - 1)

$$\frac{x \left(2 - 2 x\right)}{\left(x - 1\right)^{4}} + \frac{1}{\left(x - 1\right)^{2}}$$

Simplify

The second derivative [src]

  /      3*x  \
2*|-2 + ------|
  \     -1 + x/
---------------
           3   
   (-1 + x)

$$\frac{2 \left(\frac{3 x}{x - 1} - 2\right)}{\left(x - 1\right)^{3}}$$

Simplify

The third derivative [src]

  /     4*x  \
6*|3 - ------|
  \    -1 + x/
--------------
          4   
  (-1 + x)

$$\frac{6 \left(- \frac{4 x}{x - 1} + 3\right)}{\left(x - 1\right)^{4}}$$

Simplify

The graph

Derivative of x/(x-1)^2